%I #13 Mar 12 2021 22:24:48

%S 1,1,0,1,1,2,2,1,1,0,2,1,0,0,1,2,0,1,1,2,3,1,1,1,0,2,1,1,1,1,0,0,2,1,

%T 1,0,1,0,1,1,3,1,2,1,0,4,0,1,1,2,1,0,1,1,1,2,0,1,0,1,2,0,1,1,1,0,1,1,

%U 0,0,3,2,1,1,2,2,1,1,2,0,2,0,1,2,2,2,0

%N Expansion of f(x, x^3) * f(x^4, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

%H G. C. Greubel, <a href="/A281492/b281492.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F f(a,b) = 1 + Sum_{k=1..oo} (ab)^(k(k-1)/2)*(a^k+b^k). - _N. J. A. Sloane_, Jan 30 2017

%F Euler transform of period 18 sequence [1, -1, 1, 0, 2, -1, 1, -2, 0, -2, 1, -1, 2, 0, 1, -1, 1, -2, ...].

%F G.f.: (Sum_{k>0} x^(k*(k - 1)/2)) * (Sum_{k in Z} x^(k*(9*k + 1)/2)).

%F G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)) * (1 + x^(9*k-5)) * (1 + x^(9*k-4)) * (1 - x^(9*k)).

%F 2 * a(n) = A281451(128*n + 17).

%e G.f. = 1 + x + x^3 + x^4 + 2*x^5 + 2*x^6 + x^7 + x^8 + 2*x^10 + x^11 + ...

%e G.f. = q^5 + q^41 + q^113 + q^149 + 2*q^185 + 2*q^221 + q^257 + q^293 + ...

%t a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ -x^4, x^9] QPochhammer[ -x^5, x^9] QPochhammer[ x^9], {x, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, sumdiv(36*n + 5, d, kronecker(-4, d)) / 2)};

%Y Cf. A281451.

%K nonn

%O 0,6

%A _Michael Somos_, Jan 29 2017